Question

How do you factor $6{{x}^{2}}+14x-12$ ?

Answer 1

Hint: We are given $6{{x}^{2}}+14x-12$, we are asked to find the factor form of this, to do so we will first understand the type of equation we have, once we get that we will find the greatest common factor from each term then in the remaining term be factor using the middle term.
We use $a\times b$ in such a way that its sum or difference from the ‘b’ of the equation $a{{x}^{2}}+bx+c$ .
Once we split that we will unite all factors of the equation and we get our required answer.

Complete step by step answer:
We are given $6{{x}^{2}}+14x-12$, we are asked to find the factor of it.
To find the factor of the equation, we should see that as the highest power is 2 so it is or 2 degree polynomial. So it is a quadratic equation.
Now, to factor$6{{x}^{2}}+14x-12$, we will first find the possible greatest common factor of all these.
In 6, 14, and 12, we can see that 2 is the only possible term that can be separated, so we get –
$6{{x}^{2}}+14x-12=2\left( 3{{x}^{2}}+7x-6 \right)$
Now we will use the middle term to split
In middle term split apply on $a{{x}^{2}}+bx+c$ , we product ‘a’ by ‘c’ and then factor ‘ac’ in such a way that if product is ‘ac’ while if sum or difference made up to ‘b’.
Now, we have middle term split on $3{{x}^{2}}+7x-6$
We have $a=3,b=+7\text{ and }c=-6$
So,
$a\times c=3\times -6=-18$
Now we can see that $9\times -2=-18$ and also $9-2=7$
So we use this to split the middle term.
So,
$\begin{align}
  & 3{{x}^{2}}+7x-6 \\
 & =3{{x}^{2}}+\left( 9-2 \right)x-6 \\
 & =3{{x}^{2}}+9x-2x-6 \\
\end{align}$
We take common in the first 2 terms and the last 2 terms. So we get –
$\begin{align}
  & =3{{x}^{2}}+9x-2x-6 \\
 & =3x\left( x+3 \right)-2\left( x+3 \right) \\
\end{align}$
As $x+3$ is same, so we get –
$=\left( 3x-2 \right)\left( x+3 \right)$
So, we get –
$3{{x}^{2}}+7x-6=\left( 3x-2 \right)\left( x+3 \right)$
Now we get that –
$\begin{align}
  & 6{{x}^{2}}+14x-12=2\left( 3{{x}^{2}}+7x-6 \right) \\
 & =2\left( 3x-2 \right)\left( x+3 \right) \\
\end{align}$

So, factor form of $6{{x}^{2}}+14x-12$ is $2\left( 3x-2 \right)\left( x+3 \right)$

Note: While find the middle term using factor of $a\times c$ , we need to keep in mind that when the sign of ‘a’ and ‘c’ are same then ‘b’ is obtained by addition only, if the sign of ‘a’ and ‘c’ are different then ‘b’ can be obtained using only subtraction.
So, as we have $a=3$ and $c=-6$ different signs so ‘b’ is obtained as $9-2=7$ using subtraction.